# Minus sign differential operators

Why is the Poisson equation often written as: $-\Delta u=f$ ? Similarily, the Sturm equation is often written as: $-(pu')'+qu=f$. What purpose does the minus sign serve? Thanks

• the minus sign is to make the operator selfadjoint and so all the eigenvalues are always real and positive as a counter example imagine $\frac{d^{2}f}{dx}= \lambda_{n}f(x)$ with conditions $y(0)=y(1)=0$ are the eigenvalues real – Jose Garcia Feb 3 '14 at 12:59

For example, in the Poisson equation, assume $u\in H^1(\Omega)$, where $\Omega$ is the domain of equation and $H$ donate classical Sobolev space. We can obtain the weak form of the equation by the following deduction:
\begin{equation} (-\Delta u,v)\stackrel{\mathrm{Green's\ identity}}{=}(\nabla u,\nabla v)=(f,v) \quad \forall v\in H^1_0(\Omega) \end{equation} Where $(u,v)=\int_\Omega u v d\sigma$ is the classical $L^2$ inner product, and $v$ is the test function. In this deduction, the minus sign have been cancel by using Green's identity. Thus, the weak form of this equation have a pretty statement.